A microscopic approach to the cross diffusivity in a binary fluid mixture has been developed using the theoretical framework of continued fraction method. A suitable transformation of the velocities of distinct particles is used to formulate a continued fraction approach for the cross velocity correlation function. The self diffusivities needed for this calculation are obtained through mode coupling theory. The proposed theory is applied to a Lennard-Jones fluid mixture and the calculated cross diffusivities are found to be in good agreement with the available computer simulation results. The theory predicts the correct trend for the variation of cross diffusivity with mass and composition and also explains qualitatively the nature of the time dependence of the cross velocity correlation.